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Tue, 17 Oct 2000 00:00:00 +0200Tue, 17 Oct 2000 00:00:00 +0200Boltzmann Simulations with Axisymmetric Geometry
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/721
The paper presents theoretical and numerical investigations on simulation methods for the Boltzmann equation with axisymmetric geometry. The main task is to reduce the computational effort by taking advantage of the symmetry in the solution of the Boltzmann equation.; The reduction automatically leads to the concept of weighting functions for the radial space coordinate and therefore to a modified Boltzmann equation. Consequently the classical simulation methods have to be modified according to the new equation.; The numerical results shown in this paper - rarefied gas flows around a body with axisymmetric geometry - were done in the framework of the European space project HERMES.Jens Struckmeier; Konrad Steinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/721Tue, 17 Oct 2000 00:00:00 +0200A Comparison of Simulation Methods for Rarefied Gas Flows
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/730
Simulation methods like DSMC are an efficient tool to compute rarefied gas flows. Using supercomputers it is possible to include various real gas effects like vibrational energies or chemical reactions in a gas mixture. Nevertheless it is still necessary to improve the accuracy of the current simulation methods in order to reduce the computational effort. To support this task the paper presents a comparison of the classical DSMC method with the so called finite Pointset Method. This new approach was developed during several years in the framework of the European space project HERMES. The comparison given in the paper is based on two different testcases: a spatially homogeneous relaxation problem and a 2-dimensional axisymmetric flow problem at high Mach numbers.Jens Struckmeier; Konrad Steinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/730Tue, 17 Oct 2000 00:00:00 +0200Consistency analysis of mesh-free methods for conservation laws
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1112
Based on general partitions of unity and standard numerical flux functions, a class of mesh-free methods for conservation laws is derived. A Lax-Wendroff type consistency analysis is carried out for the general case of moving partition functions. The analysis leads to a set of conditions which are checked for the finite volume particle method FVPM. As a by-product, classical finite volume schemes are recovered in the approach for special choices of the partition of unity.Michael Junk; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1112Mon, 14 Aug 2000 00:00:00 +0200On the Efficiency of Simulation Methods for the Boltzmann Equation on Parallel Computers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/697
The paper presents a parallelization technique for the finite pointset method, a numerical method for rarefied gas flows.; First we give a short introduction to the Boltzmann equation, which describes the behaviour of rarefied gas flows. The basic ideas of the finite pointset method are presented and a strategy to parallelize the algorithm will be explained. It is shown that a static processor partition leads to an insufficient load-balance of the processors. Therefore an optimized parallelization technique based on an adaptive processor partition will be introduced, which improves the efficiency of the simulation code over the whole region of interesting flow situation. Finally we present a comparison of the CPU-times between a parallel computer and a vector computer.Jens Struckmeier; Franz-Josef Pfreundtpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/697Sun, 09 Jul 2000 00:00:00 +0200A steady-state particle method for the Boltzmann equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/624
We present a particle method for the numerical simulation of boundary value problems for the steady-state Boltzmann equation. Referring to some recent results concerning steady-state schemes, the current approach may be used for multi-dimensional problems, where the collision scattering kernel is not restricted to Maxwellian molecules. The efficiency of the new approach is demonstrated by some numerical results obtained from simulations for the (two-dimensional) BEnard's instability in a rarefied gas flow.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/624Sun, 25 Jun 2000 00:00:00 +0200Note: Alternative Discretization Schemes in Lattice Boltzmann Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/639
Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/639Sun, 25 Jun 2000 00:00:00 +0200On a Kinetic Model for Shallow Water Waves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/528
The system of shallow water waves is one of the classical examples for nonlinear, twodimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter e which leads for e to 0 to the system of shallow water waves. The corresponding equilibrium distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of nonlinear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/528Mon, 03 Apr 2000 00:00:00 +0200Generation of Random Variates Using Asymptotic Expansions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/535
Monte-Carlo methods are widely used numerical tools in various fields of application, like rarefied gas dynamics, vacuum technology, stellar dynamics or nuclear physics. A central part in all applications is the generation of random variates according to a given probability law. Fundamental techniques to generate non-uniform random variates are the inversion principle or the acceptance-rejection method. Both procedures can be quite time-consuming if the given probability law has a complicated structure.; In this paper we consider probability laws depending on a small parameter and investigate the use of asmptotic expansions to generate random variates. The results given in the paper are restrictedto first order expansions. We show error estimates for the discrepancy as well as for the bounded Lipschitz distance of the asymptotic expansion. Furthermore the integration error for some special classes of functions is given. The efficiency of the method is proved by a numerical example from rarefied gas flows.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/535Mon, 03 Apr 2000 00:00:00 +0200Tutorial on Asymptotic Analysis I
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/536
This text summarizes parts of the exercises of the tutorialon 'Asymptotic Analysis' held in the winter term 1993/94 atthe University of Kaiserslautern.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/536Mon, 03 Apr 2000 00:00:00 +0200Boltzmann Simulation by Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/541
Particle methods to simulate rarefied gas flows have found an increasing interest in Computational Fluid Dynamics during the last decade, see for example [1], [2], [3] and [4]. The general goal is to develop numerical schemes which are reliable enough to substitute real windtunnel experiments, needed for example in space research, by computer experiments. In order to achieve this goal one needs numerical methods solving the Boltzmann equation including all important physical effects. In general this means 3D computations for a chemically reacting rarefied gas. With codes of this kind at hand, Boltzmann simulation becomes a powerful tool in studying rarefied gas phenomena.Helmut Neunzert; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/541Mon, 03 Apr 2000 00:00:00 +0200Implicit and Iterative Methods for the Boltzmann Equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/553
The paper presents some approximation methods for the Boltzmann equation. In the first part fully implicit discretization techniques for the spatially homogeneous Boltzmann equation are investigated. The implicit equation is solved using an iteration process. It is shown that the iteration converges to the correct solution for the moments of the distribution function as long as the mass conservation is strictly fulfilled. For a simple model Boltzmann equation some unexpected features of the implicit scheme and the corresponding iteration process are clarified. In the second part a new iteration algorithm is proposed which should be used for the stationary Boltzmann equation. The realization of the method is very similar to the standard splitting algorithms except some new stochastic elements.A.V. Bobylev; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/553Mon, 03 Apr 2000 00:00:00 +0200Second Order Scheme for the Spatially Homogeneous Boltzmann Equation with Maxwellian Molecules
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/557
In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction of the discretization parameter as well as on the differential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit particle scheme for the Boltzmann equation with Maxwellian molecules. The present paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second order particle method, when using an equiweighting of explicit and implicit discretization.Jens Struckmeier; Konrad Steinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/557Mon, 03 Apr 2000 00:00:00 +0200Numerical Simulation of the Stationary One-Dimensional Boltzmann Equation by Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/558
The paper presents a numerical simulation technique - based on the well-known particle methods - for the stationary, one-dimensional Boltzmann equation for Maxwellian molecules. In contrast to the standard splitting methods, where one works with the instationary equation, the current approach simulates the direct solution of the stationary problem. The model problem investigated is the heat transfer between two parallel plates in the rarefied gas regime. An iteration process is introduced which leads to the stationary solution of the exact - space discretized - Boltzmann equation, in the sense of weak convergence.Jens Struckmeier; A.V. Bobylevpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/558Mon, 03 Apr 2000 00:00:00 +0200Simulation of Boundary Value Problems for the Boltzmann Equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/580
The paper presents numerical results on the simulation of boundary value problems for the Boltzmann equation in one and two dimensions. In the one-dimensional case, we use prescribed fluxes at the left and diffusive conditions on the right end of a slab to study the resulting steady state solution. Moreover, we compute the numerical density function in velocity space and compare the result with the Chapman-Enskog distribution obtained in the limit for continuous media. The aim of the two-dimensional simulations is to investigate the possibility of a symmetry break in the numerical solution.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/580Mon, 03 Apr 2000 00:00:00 +0200Particle Methods: Theory and Applications
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/586
In the present paper a review on particle methods and their applications to evolution equations is given. In particular, particle methods for Euler- and Boltzmann equations are considered.Helmut Neunzert; Axel Klar; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/586Mon, 03 Apr 2000 00:00:00 +0200Some Estimates on the Boltzmann Collision Operator
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/599
The paper presents some new estimates on the gain term of the Boltzmann collision operator. For Maxwellian molecules, it is shown that the L -norm of the gain term can be bounded in terms of the L1 and L -norm of the density function f. In the case of more general collision kernels, like the hard-sphere interaction potential, the gain term is estimated pointwise by the L -norm of the density function and the loss term of the Boltzmann collision operator.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/599Mon, 03 Apr 2000 00:00:00 +0200Adaptive Load Balance Techniques in Parallel Rarefied Gas Simulations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/601
The paper presents some adaptive load balance techniques for the simulation of rarefied gas flows on parallel computers. It is shown that a static load balance is insufficient to obtain a scalable parallel efficiency. Hence, two adaptive techniques are investigated which are based on simple algorithms. Numerical results show that using heuristic techniques one can achieve a sufficiently high efficiency over a wide range of different hardware platforms.S. Antonov; Franz-Josef Pfreundt; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/601Mon, 03 Apr 2000 00:00:00 +0200Particle Methods for Evolution Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/602
Michael Junk; Axel Klar; Jens Struckmeier; Sudarshan Tiwaripreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/602Mon, 03 Apr 2000 00:00:00 +0200Particle Methods in Fluid Dynamics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/621
Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/621Mon, 03 Apr 2000 00:00:00 +0200The Rayleigh-Benard Convection in Rarefied Gases
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/629
In the present paper we investigate the Rayleigh-Benard convection in rarefied gases and demonstrate by numerical experiments the transition from purely thermal conduction to a natural convective flow for a large range of Knudsen numbers from 0.02 downto 0.001. We address to the problem how the critical value for the Rayleigh number defined for incompressible vsicous flows may be translated to rarefied gas flows. Moreover, the simulations obtained for a Knudsen number Kn=0.001 and Froude number Fr=1 show a further transition from regular Rayleigh-Benard cells to a pure unsteady behavious with moving vortices.Carlo Cercignani; Maria Cristina Giurin; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/629Mon, 03 Apr 2000 00:00:00 +0200Transition from Kinetic Theory to Macroscopic Fluid Equations: A Problem fo Domain Decomposition and a Source for New Algorithms
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/636
In the paper we discuss the transition from kinetic theory to macroscopic fluid equations, where the macroscopic equations are defined as aymptotic limits of a kinetic equation. This relation can be used to derive computationally efficient domain decomposition schemes for the simulaion of rarefied gas flows close to the continuum limit. Moreover, we present some basic ideas for the derivation of kinetic induced numerical schemes for macroscopic equations, namely kinetic schemes for general conservation laws as well as Lattice-Boltzmann methods for the incompressible Navier-Stokes equations.Axel Klar; Helmut Neunzert; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/636Mon, 03 Apr 2000 00:00:00 +0200Applications of a New Model for the Differential Cross-Section of a Classical Polyatomic Gas
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/643
We give a comparison of various differential cross-section models for a classical polyatomic gas for a homogeneous relaxation problem and the shock wave profiles at Mach numbers 1.71 and 12.9. Besides the standard Borgnakke-Larsen model and its generalizations to an energy dependent coefficient to control the amnount of rotationally elastic and completely inelastic collisions, we discuss some new models recently proposed by the same authors. Moreover, we present numerical algorithms to implement the models in a particle or Monte-Carlo code and compare the numerical shock wave profiles with existing experimental data.Carlo Cercignani; Maria Lampis; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/643Mon, 03 Apr 2000 00:00:00 +0200A Singular-Perturbed Two-Phase Stefan Problem Due to Slow Diffusion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/648
The asymptotic behaviour of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.Jens Struckmeier; Andreas Unterreiterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/648Mon, 03 Apr 2000 00:00:00 +0200Fast Generation of Low-Discrepancy Sequences
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/732
The paper presents a fast implementation of a constructive method to generate a special class of low-discrepancy sequences which are based on Van Neumann-Kakutani tranformations. Such sequences can be used in various simulation codes where it is necessary to generate a certain number of uniformly distributed random numbers on the unit interval.; From a theoretical point of view the uniformity of a sequence is measured in terms of the discrepancy which is a special distance between a finite set of points and the uniform distribution on the unit interval.; Numerical results are given on the cost efficiency of different generators on different hardware architectures as well as on the corresponding uniformity of the sequences. As an example for the efficient use of low-discrepancy sequences in a complex simulation code results are presented for the simulation of a hypersonic rarefied gas flow.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/732Mon, 03 Apr 2000 00:00:00 +0200On a Recursive Approximation of Singularity Perturbed Parabolic Equations (extended version)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/653
The asymptotic analysis of IBVPs for the singularly perturbed parabolic PDE ... in the limit epsilon to zero motivate investigations of certain recursively defined approximative series ("ping-pong expansions"). The recursion formulae rely on operators assigning to a boundary condition at the left or the right boundary a solution of the parabolic PDE. Sufficient conditions for uniform convergence of ping-pong expansions are derived and a detailed analysis for the model problem ... is given.Jens Struckmeier; Andreas Unterreiterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/653Wed, 25 Aug 1999 00:00:00 +0200